1 NORMED ALGEBRAS 305 spectrum and Gelfand transformation (15.3), have found many other applica- tions in modern analysis, notably in the theory of analytic functions. Some of these applications are touched on in the problems, and the reader interested in this aspect is referred to [35] and [29]. The central part of this chapter is the study of representations of algebras with involution, which enables such an algebra, given "abstractly," to be "realized" as an algebra of operators on a Hilbert space. The essential notion in the modern development of this theory is that of a Hilbert form, which is closely related to that of a Hilbert algebra (15.7). We shall study in detail only two particular aspects of the theory of Hilbert algebras: the first (15.8) prepares the way for the theory of representations of compact groups (Chapter XXI), and the second (15.9) for spectral theory and harmonic analysis (Chapter XXII). The reader who wishes to go further (notably in view of the deep and difficult theory of representations of locally compact groups) is warmly recommended to read the two beautiful volumes by J. Dixmier ([24] and [25]) which dominate the subject. The Hilbert spectral theory (15.10 and 15.11) appears in our treatment as an immediate particular case of the general theorem of Bochner-Godement (15.9). It can be reached more directly and rapidly (Section 15.10, Problem 2) from the Gelfand-Neumark theorem (15.4), but it seemed to us to be more instructive to deduce it from a much more powerful theorem which is the cornerstone of harmonic analysis, even if this requires a small additional effort. The applications of spectral theory are not limited to those referred to above. Among the most celebrated we should mention at least the following: (1) one of the most elegant theories in Analysis, namely the "moment problem" inaugurated by Stieltjes, with its many ramifications (analytic functions, orthogonal polynomials, Jacobi matrices, continued fractions, etc), which fits admirably into the theory of unbounded Hermitian operators; (2) the interesting relations between ergodic theory and spectral theory; (3) perturbation theory. Some of the important results of these theories are mentioned in the problems, and the reader is referred for more ample informa- tion to the works [20], [28], [30], and [32] in the References. 1. NORMED ALGEBRAS When we speak of algebras in this chapter, we shall always mean algebras over the field of complex numbers C. A normed algebra is defined to be an algebra A endowed with a norm JCH* \\x\\ (5.1) satisfying the inequality (15.1.1) \\xy\\ ^ \\x\\ • \\y\\ for all x, y in A.how that as y